Comments on the California Modeling Appendix

The draft Framework has a place where you can submit comments, but each box is limited to 500 characters. I know I go on too long, but seriously, I can’t fit what I have to say in that space!

So, California Modeling Persons, I’m really interested in what gets said about modeling in the Framework. I’d love to help this important topic make its way into the schools in a way that will not make teachers, parents, or Mathematically Correct roll their eyes in despair. I’m worried that if we proceed without great care and thoughtfulness, modeling will become a word-du-jour about which an army of consultants will make powerpoint slides, and which will then fade away by the time the next set of standards or frameworks emerges.

Comments close in two days (and I have no actual job that will justify my writing about this) so I apologize that I will not be creating a fully-formed and coherent narrative suitable for cutting and pasting, but will only be making comments. Besides, Usiskin and others have written well on this subject.

So here we go:

First, be sure to read Lesh on Model-Eliciting Activities (MEAs). You will find references here and a longer summary here.

Now, references to actual line numbers in the April 2013 draft. Do not consider these as editing suggestions at that line, but rather springboards for considering larger issues. These might (I hope) result in changes elsewhere in the document.

Line 9.5: What isn’t mathematical modeling. Excellent, although I disagree that it has to start in the real world. More discussion below.

Line 37: “How could we measure that?” I think the document could make more of measurement. In general, measurement gets short shrift in standards documents. Yet measurement—especially indirect measurement—is an important component of modeling.

Line 56: “This is the step of creating a mathematical model.” Here is where I have my first big problem, one I have been grappling with in recent posts. If you have time, please review them. Here’s the nutshell: I agree that modeling, in the modeling-cycle diagram, certainly occurs when you cross the reality-mathematics barrier. It’s when you mathematize the real world. The peanut-butter cup (line 300) is a great example.

But I think modeling also occurs in another place, one that stats teachers will recognize: If you have a collection of data, and you try to find a function that represents that data fairly, you’re modeling. In fact, modeling data with functions is a really important part of modeling, and one of the places where students can connect the real world to the math (i.e., elementary functions) that’s part of their syllabus. And this does not occur as you cross that reality-math barrier, but rather when you make the transition from already-mathematized data (e.g., measurements of a real-world phenomenon) to a more abstract and symbolic simplification and generalization.

It’s as if there’s a second modeling cycle inside the cycle: in one modeling step, we simplify reality into quantitative data. Then, in another step, we simplify (and generalize) data by approximating it with a function. 

I’ve indented that because I think it’s one of the most important things I have to say. I know, this may be a pointy-headed, Jesuit-worthy fine distinction that needn’t go further. But it’s an example of a place where simple definitions don’t work well, and where it would be easy for teachers and developers to lose some of their grasp of what’s modeling and what’s not. (The opening this-is-not-modeling box is a great help). It also points out that modeling-data-with-functions may be a different genre of modeling that deserves its own approach. Are the skills, knowledge, predispositions, and habits of mind students need for this kind of modeling (finding the function) the same as those they need for, say, geometrical modeling (approximating the peanut-butter cup as embedded cylinders)? This would require more thought.

Line 13: “Put simply, mathematical modeling is the process of using mathematical tools and methods to ask and answer questions about real world situations (Abrams, 2012)“. This is mostly true, but I actually disagree. We can do modeling about abstract, pure-math topics as well. (See this post) So I worry about this being the definition. It both steers people away from pure-math examples of modeling, and drives those who only read to line 14 towards the mistaken notion that anything real-world is modeling.

I prefer thinking of the essence of modeling as simplification, as in:

A model is an abstract, simplified, and idealized representation of a real object, a system of relations, or an evolutionary process, within a description of reality. (Henry, 2001, p. 151; quoted in Chaput et al., 2008)

Line 74: “[in contrast to a word problem]…often in a modeling situation the exact solution path is unclear.” I agree, but let’s not make the contrast too stark. Way back in the olden days—was it the 1985 Framework? 1992? We made a big deal of the distinction between exercises, problems, and investigations. One of the essential aspects of a problem was that the exact solution path was unclear. If it was clear, it was an exercise; we claimed that most word problems were not problems at all. I agree that modeling tasks have this property, but so do good problems of all sorts.

Line 82: The paragraph continues to say, for example, that modeling problems are “…multifaceted and multidisciplinary: students’ final products encompass a variety of representational formats…” This is reminiscent of Lesh MEAs. I agree, but saying it this way so early in the document may lead readers to believe that modeling tasks are identical to projects. Much later in the document, you make some important points, especially that there are lots of modeling tasks students at all levels can do that may encompass the whole modeling cycle but are nevertheless (a) modeling and (b) useful.

Line 93: “When modeling is seen as a vehicle for teaching mathematics, emphasis is not placed on students becoming proficient modelers themselves.” I see what you mean, but I’d be happier if the distinction (with the following paragraph) were more like this:

“It is possible—even desirable—to introduce and refine mathematical topics in contexts that involve modeling. That way, in a math-focused task, students get a meaningful setting and experience with modeling skills. A can-optimization task, for example, would give students the chance to find and work with formulas for the surface area and volume of a cylinder, and decide what to hold constant, but it might tell students to approximate the can as a cylinder—rather than letting them decide for themselves.”

line 136: section: The Role of the Teacher: I agree with all of this; my problem is that I don’t see this as unique to modeling. Neither do you! (see lines 266–268) We do want students to come up with their own questions, and we have asked for decades that teachers become more like consultants. This section should acknowledge that right up front. The point is that modeling tasks more naturally lend themselves to this sort of teaching. It’s (fortunately) harder to imagine how to get them to work at all using direct, teacher-centered instruction. Teachers can use the advent of modeling as an opportunity to improve our practice, and learn how to make these things fly—not just with modeling but with all sorts of tasks.

What you don’t want, as a writer, is for the most advanced, progressive readers to come to this section and say to themselves, this is old news. I should stop reading now.

Line 166: It would be great to see what this graphic looks like. As I have written elsewhere, I’m naturally chary of something called “four steps to solve a modelling task.” (two l’s because they’re Not From Here.)

Line 204: The Abrams table: terrific. Tables of contrasts and distinctions work for me. Notice also that the contrast is with mathematical exercises, so doesn’t raise the problem of conflating modeling with problem-solving.

Line 251: “Consider Usiskin’s ‘reverse given-find’ problems.” Great problems! But are they modeling? I don’t think so. No real-world (not so important for me) and no simplification (which is crucial). If we argue that these are modeling, it will get too easy to argue that any rich task is modeling. Then anything good is modeling. And from there it’s an easy step to anything is modeling. Sounds flip, but I’m serious. We MUST be careful about this.

Line 309: “…the modeling process is seldom linear…” Okay, one copyedit comment: when you’ve been talking about functions, avoid using linear in a different meaning. The reader is already weary; you don’t want him or her to think, even for a moment, that to be modeling, a relationship has to be curved.

Line 351: “…apply particular mathematical content at a given grade level.” Thank you. This comes too late for my taste; the grade-level caveat is important. I think that although mathematical modeling involves using math to simplify and represent reality, to count as actually learning about modeling, the math in question needs to have some relationship to the grade level. High-school students may indeed compare groups based on proportions of some attribute (more girls than boys prefer soccer to basketball, say), but deciding on proportion, in itself, though critical and often difficult, doesn’t count (for me) as modeling. Why not? Because using proportions to compare groups is supposed to be a middle-grades skill. High-school students can’t do it reliably, I know. But somehow I don’t think high-school use of proportion, in itself, meets the standard.

General Comments

In addition to bringing some important ideas forward, and perhaps rethinking the way some ideas are presented, the document needs some careful tightening. Too often I found myself thinking that we were repeating ourselves. Two examples are the notion that teaching will have to change and how; and the idea that modeling tasks often incorporate multiple standards. These ideas are important, but we should be more careful about how they are repeatedly introduced. (And we should look for other unfortunate repeats.)

Then, the modeling course. It’s worth pointing out that this is (only) a (partial) list of topics. It’s too easy to look at the table as an outline, imagining that one would cover exponentials before quadratics (for example). But it’s important, in general, not to have separate chapters for separate functions!

I also have some quibbles with individual boxes. Some are too limiting. Exponentials, for example, arise in geometrical contexts as well; see this post for an example.

A bigger problem is that although the table acknowledges lots of different functions, there’s no mention about how to work with them. In a modeling course, I’d be sure to have understanding transformations as one of my goals. Another would be coping with variability.

In general, this modeling draft comes from a relatively data-poor perspective. It does not attend very much to the role of data and emerging principles of data science in mathematical modeling. It leans towards applied math rather than data analysis. It’s all good stuff, don’t get me wrong. But I think that working with data was seldom in the forefront of the writer’s mind.


Chaput, B., Girard, J., & Henry, M. (2008). Modeling and simulation in statistics education. Proceedings of the ICMI Study 18 and 2008 IASE Round Table Conference.


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Tim Erickson

Math-science ed freelancer and sometime math teacher. In 2014–15, at Mills College in Oakland, California.

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